The mathematical theory of integrable systems has been
described as one of the most profound advances of twentieth century
mathematics. They lie at the boundary of mathematics and physics and were
discovered through a famous paradox that arises in a model devised to
describe the thermal properties of metals (called the Fermi-Pasta-Ulam
paradox).

In attempting to resolve this paradox, Kruskal and Zabusky discovered
exceptional properties in the solutions of a non-linear PDE, called the
Korteweg-de Vries equation (KdV). These properties showed that although the
solutions are waves, they interact with each other as though they were
particles, i.e., without losing their shape or speed, until then thought to
be impossible for solutions of non-linear PDEs. Kruskal invented the name
solitons for these solutions. Here is a picture of two solitons
interacting.

Solitons are known to arise in other non-linear PDEs and also in partial difference equations. These systems and their symmetry reductions are now called integrable systems. These systems occur as universal limiting models in many physical situations.

This course introduces the mathematical properties of such systems. In particular, we will study their solutions, symmetry reductions called the Painlevé equations and their discrete versions. It focuses on mathematical methods created to describe the solutions of such equations and their interrelationships. More details about the course, including course objectives and outcomes and details about the assessment and exam can be read on the Information Sheet (PDF).